Why scale tensor products like that?

Studying the sum of angular momentum in quantum mechanics, tensor products were introduced to us to get the general system from the individual states. There’s a property that says: $(av)otimes w = a(v otimes w)= votimes (aw) $. We will call it (1).
But I don’t get it’s physical meaning. It’s as if I scale one vector, but not the other, and the vector of the total system scales by that factor¿?.

For example, for two particles with spin $frac{1}{2}$, I have the state $ |frac{1}{2},m_1rangle otimes |frac{1}{2},m_2rangle $ for $m_i=frac{1}{2},-frac{1}{2}$. If I measure the $z$ component of the spin of the first particle ($S_1$), I get the value $hbar m_1$, with $|frac{1}{2},m1rangle$ an eigenvector of that operator. Hence, we have that: $ (S_{1,z} otimes 1) (|frac{1}{2},m_1rangle otimes |frac{1}{2},m_2rangle)=(hbar m_1 |frac{1}{2},m_1rangle) otimes |frac{1}{2},m_2rangle $ and by the property (1) that’s just equal to: $ hbar m_1( |frac{1}{2},m_1rangle otimes |frac{1}{2},m_2rangle)$. Which seems to imply that $|frac{1}{2},m_1rangle otimes |frac{1}{2},m_2rangle$ is an eigenvector of $S_{1,z} otimes 1$ with eigenvalue $hbar m_1$ (the measurement of state 1 is scaling the total system by that factor). Or what’s worse, by the third equality in (1), that: $ (S_{1,z} otimes 1) (|frac{1}{2},m_1rangle otimes |frac{1}{2},m_2rangle)= |frac{1}{2},m_1rangle otimes (hbar m_1 |frac{1}{2},m_2rangle) $ (that the measurement in the first state scales the second one by that factor ¿¿¿???).
But that’s not the case in reality. The measurement in state 1 doesn’t change things that way.

I don’t get what’s going on, I don’t understand much tensor products. If you can shed some light on this.